Geodetic number of cyclic graphs

A vertex subset [Formula: see text] of a graph [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] is in some [Formula: see text] geodesic for any [Formula: see text]. The minimum cardinality of such a set is the geodetic number, which is denoted as [Formula: see text]. The geodetic number has wide applications in location-allocation problems and most of the networks which are used in this study are cyclic. Therefore, the study of bounds of the geodetic number of cyclic graphs has significance. In this paper, we discuss the upper and lower bounds of the geodetic and restrained geodetic numbers of some families of trees and cyclic graphs. We also characterize graphs that possess the same value of geodetic and restrained geodetic numbers among trees and unicyclic graphs. Finally, we propose a conjecture on the characterization of graphs that attain equality in geodetic and restrained geodetic numbers among cacti.